- Previous Article
- JCD Home
- This Issue
-
Next Article
Rigorous enclosures of rotation numbers by interval methods
On the computation of attractors for delay differential equations
1. | Institute for Mathematics, University of Paderborn, D-33095 Paderborn |
2. | Department of Mathematics, Paderborn University, 33095 Paderborn, Germany, Germany |
References:
[1] |
A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.
doi: 10.1016/0167-2789(85)90093-4. |
[2] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).
|
[3] |
C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.
doi: 10.1016/S0022-0396(02)00148-1. |
[4] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).
doi: 10.1007/978-1-4612-3506-4. |
[5] |
J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.
doi: 10.1016/0167-2789(84)90275-6. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[7] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.
doi: 10.1007/s002110050240. |
[8] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[9] |
M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).
doi: 10.1103/PhysRevLett.94.231102. |
[10] |
R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.
doi: 10.1137/1010058. |
[11] |
J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.
doi: 10.2140/pjm.1951.1.353. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957). Google Scholar |
[13] |
J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.
doi: 10.1016/0167-2789(82)90042-2. |
[14] |
C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[15] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.
doi: 10.1137/S106482750238911X. |
[16] |
G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69. Google Scholar |
[17] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.
doi: 10.1016/j.physd.2010.03.009. |
[18] |
C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.
doi: 10.1137/040608295. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[20] |
B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.
doi: 10.1088/0951-7715/12/5/303. |
[21] |
B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.
doi: 10.1063/1.166450. |
[22] |
I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.
doi: 10.1016/j.physd.2004.04.004. |
[23] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.
doi: 10.1126/science.267326. |
[24] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[25] |
J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.
doi: 10.1088/0951-7715/18/5/013. |
[26] |
T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.
doi: 10.1137/080718772. |
[27] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.
doi: 10.1007/BF01053745. |
[28] |
C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.
|
[29] |
J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.
doi: 10.1007/s003329900072. |
[30] |
F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.
|
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.
doi: 10.1007/978-1-4612-0645-3. |
[32] |
C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.
doi: 10.1007/s10915-008-9256-y. |
[33] |
S. Willard, General Topology,, Addison-Wesley, (1970).
|
show all references
References:
[1] |
A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser, Asymptotic chaos,, Physica D: Nonlinear Phenomena, 14 (1985), 327.
doi: 10.1016/0167-2789(85)90093-4. |
[2] |
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Oxford University Press, (2013).
|
[3] |
C. Chicone, Inertial and slow manifolds for delay equations with small delays,, Journal of Differential Equations, 190 (2003), 364.
doi: 10.1016/S0022-0396(02)00148-1. |
[4] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,, Springer-Verlag, (1989).
doi: 10.1007/978-1-4612-3506-4. |
[5] |
J. D. Crawford and S. Omohundro, On the global structure of period doubling flows,, Physica D: Nonlinear Phenomena, 13 (1984), 161.
doi: 10.1016/0167-2789(84)90275-6. |
[6] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - Set oriented numerical methods for dynamical systems,, in Ergodic Theory, (2001), 145.
|
[7] |
M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors,, Numerische Mathematik, 75 (1997), 293.
doi: 10.1007/s002110050240. |
[8] |
M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM Journal on Numerical Analysis, 36 (1999), 491.
doi: 10.1137/S0036142996313002. |
[9] |
M. Dellnitz, O. Junge, M. Lo, J. E. Marsden, K. Padberg, R. Preis, S. Ross and B. Thiere, Transport of Mars-crossing asteroids from the quasi-Hilda region,, Physical Review Letters, 94 (2005).
doi: 10.1103/PhysRevLett.94.231102. |
[10] |
R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasi-linear differential equations with small delays,, SIAM Review, 10 (1968), 329.
doi: 10.1137/1010058. |
[11] |
J. Dugundji, An extension of Tietze's theorem,, Pacific J. Math., 1 (1951), 353.
doi: 10.2140/pjm.1951.1.353. |
[12] |
N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory,, Interscience Publishers, (1957). Google Scholar |
[13] |
J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system,, Physica D, 4 (1982), 366.
doi: 10.1016/0167-2789(82)90042-2. |
[14] |
C. Foias, M. Jolly, I. Kevrekidis, G. Sell and E. Titi, On the computation of inertial manifolds,, Physics Letters A, 131 (1988), 433.
doi: 10.1016/0375-9601(88)90295-2. |
[15] |
G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost invariant sets and cycles,, SIAM Journal on Scientific Computing, 24 (2003), 1839.
doi: 10.1137/S106482750238911X. |
[16] |
G. Froyland, C. Horenkamp, V. Rossi, N. Santitissadeekorn and A. Sen Gupta, Three-dimensional characterization and tracking of an Agulhas ring,, Ocean Modelling, 52 (2012), 69. Google Scholar |
[17] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems,, Physica D: Nonlinear Phenomena, 239 (2010), 1527.
doi: 10.1016/j.physd.2010.03.009. |
[18] |
C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes,, SIAM Journal on Applied Dynamical Systems, 4 (2005), 711.
doi: 10.1137/040608295. |
[19] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied mathematical sciences, (1993).
doi: 10.1007/978-1-4612-4342-7. |
[20] |
B. R. Hunt and V. Y. Kaloshin, Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces,, Nonlinearity, 12 (1999), 1263.
doi: 10.1088/0951-7715/12/5/303. |
[21] |
B. Krauskopf and H. Osinga, Two-dimensional global manifolds of vector fields,, Chaos: An Interdisciplinary Journal of Nonlinear Science, 9 (1999), 768.
doi: 10.1063/1.166450. |
[22] |
I. Kukavica and J. C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem,, Physica D: Nonlinear Phenomena, 196 (2004), 45.
doi: 10.1016/j.physd.2004.04.004. |
[23] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.
doi: 10.1126/science.267326. |
[24] |
I. Mezić and A. Banaszuk, Comparison of systems with complex behavior,, Physica D: Nonlinear Phenomena, 197 (2004), 101.
doi: 10.1016/j.physd.2004.06.015. |
[25] |
J. C. Robinson, A topological delay embedding theorem for infinite-dimensional dynamical systems,, Nonlinearity, 18 (2005), 2135.
doi: 10.1088/0951-7715/18/5/013. |
[26] |
T. Sahai and A. Vladimirsky, Numerical methods for approximating invariant manifolds of delayed systems,, SIAM J. Applied Dynamical Systems, 8 (2009), 1116.
doi: 10.1137/080718772. |
[27] |
T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Stat. Phys., 65 (1991), 579.
doi: 10.1007/BF01053745. |
[28] |
C. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems,, in Ergodic Theory, (2001), 191.
|
[29] |
J. Stark, Delay embeddings for forced systems. I. Deterministic forcing,, Journal of Nonlinear Science, 9 (1999), 255.
doi: 10.1007/s003329900072. |
[30] |
F. Takens, Detecting strange attractors in turbulence,, Springer Lecture Notes in Mathematics, 898 (1981), 366.
|
[31] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997), 978.
doi: 10.1007/978-1-4612-0645-3. |
[32] |
C. Vandekerckhove, I. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold,, Journal of Scientific Computing, 39 (2009), 167.
doi: 10.1007/s10915-008-9256-y. |
[33] |
S. Willard, General Topology,, Addison-Wesley, (1970).
|
[1] |
Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226 |
[2] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[3] |
Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021066 |
[4] |
V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 |
[5] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[6] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[7] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[8] |
Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021022 |
[9] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[10] |
Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881 |
[11] |
Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 |
[12] |
F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605 |
[13] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
[14] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[15] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
[16] |
Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 |
[17] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[18] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[19] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[20] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]